PredatorPrey Modeling
Introduction

Professor Short introduced Epidemic Models

Mathematical models showed that a disease like malaria could be extinguished
without destroying all mosquitoes

Ecological significance is that below a critical number populations can
go extinct

We examine the population dynamics of a simple predatorprey system
Ecology of PredatorPrey System
this image kindly provided by Tom
and Pat Leeson, who retain the copyright

Hudson Bay company kept careful records of all furs from the early 1800s
into the 1900s

Assume that furs are representative of the populations in the wild because
of the trapping intensity and trapping techniques

Records of furs showed distinctive oscillations with a period of about
12 years for lynx and hares

Lynx primarily eat snowshoe hares, which makes this a rare two species
(simplified) interaction  (Mathematical models have a hard time understanding
multiple species interactions)

Ecologists have predicted that in a simple predatorprey system that a
rise in prey population is followed (with a lag) by a rise in the predator
population  when the predator population is sufficiently high, then the
prey population begins dropping  thus, oscillations occur

Can a mathematical model predict this?

What causes cycles to slow or speed up? What affects the amplitude of the
oscillation or do you expect to see the oscillations damp to a stable equilibrium?

The models tend to ignore factors like climate and other complicating factors
 how significant are these?
Basic Population Model

Single species growth model with population P_{n}
and growth function g(P_{n})
is given by
P_{n}_{+1}
=
P_{n} +
g(P_{n}).

Malthusian Growth model satisfies
P_{n}_{+1}
=
P_{n} +
kP_{n} = (1 +
k)P_{n}.

This equation is easily solved (recall compound interest problems)
P_{n}
= (1 +
k)P_{n}_{1} = (1 +
k)(1 +
k)P_{n}_{2}
= (1 +
k)^{2}P_{n}_{2}
P_{n}
= (1 +
k)^{n}P_{0}

This gives exponential growth. Until recently and with a few exceptions
like the plague years, human growth has been very consistently Malthusian.

When the growth function g(P_{n})
is more complicated, then other mathematical techniques are needed. We
will use computer simulations.
PredatorPrey Model (LotkaVolterra)

Define the hare population by H_{n}
and the lynx population by L_{n}

Assume the primary growth of the hare population is Malthusian, a_{1}H_{n},
(in the absence of lynx) and that the lynx population, b_{1}L_{n},
(in the absence of hares) is negative Malthusian

Assume that the primary loss of hares is due to predation (contact) with
lynx, a_{2}H_{n}L_{n},
and that the growth of the lynx population is from energy derived from
eating hares, b_{2}H_{n}L_{n}

The LotkaVolterra model is given by
H_{n}_{+1}
=
H_{n} +
a_{1}H_{n} 
a_{2}H_{n}L_{n}
L_{n}_{+1}
=
L_{n} 
b_{1}L_{n} +
b_{2}H_{n}L_{n}
Simulation of the Model

Excel worksheet to be developed!
Model of Fishing

Following World War I, Volterra examined the fishing data for Italy and
discovered that the percent of sharks and skates in the fishing catch rose
during the years of the war
Percentages of
predators in the Fiume fish catch
1914 
1915 
1916 
1917 
1918 
1919 
1920 
1921 
1922 
1923 
12 
21 
22 
21 
36 
27 
16 
16 
15 
11 


These data don't show oscillations, but there is clearly a rise and fall
of the percent of sharks and skates in the fish catch due to effects of
the war. What caused this?

Volterra used the predatorprey model to show why this effect could be
predicted

Let F_{n} be the food fish
population and S_{n }be the
shark and skate population. To the LotkaVolterra model, we add the effects
of human fishing (using nets, a_{3}F_{n}
and  b_{3}S_{n }),
then the mathematical model becomes
F_{n}_{+1}
=
F_{n} +
a_{1}F_{n} 
a_{2}F_{n}S_{n}

a_{3}F_{n}
S_{n}_{+1}
=
S_{n} 
b_{1}S_{n} +
b_{2}F_{n}S_{n
}
b_{3}S_{n}
Equilibria

An equilibrium for a population is when the population stays the same for
all time or for each value of n. For
the previous model, we have
F_{n}_{+1}
=
F_{n
}= F_{e} and S_{n}_{+1}
=
S_{n
}= S_{e}

It can be shown (mathematically) that the average population about a cycle
of the LotkaVolterra model is its equilibrium value. Thus, if the fishing
populations are cycling less than annually, the annual catch should reflect
the equilibrium population

Substitute the equilibrium information into the equations above
F_{e}
=
F_{e} +
a_{1}F_{e} 
a_{2}F_{e}S_{e}

a_{3}F_{e}
S_{e}
=
S_{e} 
b_{1}S_{e} +
b_{2}F_{e}S_{e
}
b_{3}S_{e}

Apply some algebra. The first two terms of each equation cancel, then factor
F_{e
}from
the first equation and S_{e
}from
the second equation. The result is
F_{e }(a_{1}

a_{2}S_{e} 
a_{3}) = 0
S_{e}(b_{1}
+
b_{2}F_{e
} b_{3}) = 0

The product of two factors being zero means one of the factors is zero.
From the first equation
F_{e }=
0 or a_{1} 
a_{2}S_{e}

a_{3} = 0
S_{e }=
0 or b_{1} +
b_{2}F_{e}

b_{3} = 0

Simultaneously, solving the two equations, we have the two equilibria,
either
F_{e }=
0 and S_{e
}=
0
or
F_{e }=
(b_{1} +
b_{3})/b_{2} and
S_{e
}=
(a_{1} 
a_{3})/a_{2}
Relating Equilibria to Fishing Data

With NO fishing, the nonzero equilibrium
is
F_{e }=
b_{1}/b_{2}and
S_{e}=
a_{1}/a_{2}

With fishing, the nonzero equilibrium is
F_{e }=
(b_{1} +
b_{3})/b_{2} and
S_{e
}=
(a_{1} 
a_{3})/a_{2}

Notice that as the level of fishing increases (a_{3}
and b_{3} increasing), the
equilibrium value for the food fish increases, while the equilibrium for
the sharks and skates decreases. During World War I, the fishing fleets
would be less likely to go out. Thus, the level of fishing decreases, which
aids the equilibrium for the sharks and skates as reflected in the data!
Similar Application for the Agricultural Industry

Consider an agricultural situation, such as scale insects and lady bugs,
and the application of pesticides.

Without pesticides, the insects form a classical predatorprey situation
as modeled above.

The application of pesticides is like the situation above with the fishing
industry netting fish. The pesticide generally kills both the prey insect
(which is usually the agricultural pest) and the predator species.

Our analysis above shows that the prey species in the long run benefits
from this type of application. Thus, the agricultural pest actually does
better after application of pesticides (after a recovery time). This process
causes even larger amplitude oscillations, so nastier outbreaks.

Conclusion: The farmers and public lose
from pesticides, while the chemistry industry benefits!